A characterization of n-dimensional hypersurfaces in $R^{n+1}$ with commuting curvature operators

Yulian T. Tsankov

A characterization of n-dimensional hypersurfaces in

R^{n + 1}

with commuting curvature operators

Yulian T. Tsankov

Banach Center Publications, Tome 68 (2005), p. 205-209 / Harvested from The Polish Digital Mathematics Library

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Résumé

Let Mⁿ be a hypersurface in $R^{n + 1}$ . We prove that two classical Jacobi curvature operators $J_{x}$ and $J_{y}$ commute on Mⁿ, n > 2, for all orthonormal pairs (x,y) and for all points p ∈ M if and only if Mⁿ is a space of constant sectional curvature. Also we consider all hypersurfaces with n ≥ 4 satisfying the commutation relation $(K_{x, y} \circ K_{z, u}) (u) = (K_{z, u} \circ K_{x, y}) (u)$ , where $K_{x, y} (u) = R (x, y, u)$ , for all orthonormal tangent vectors x,y,z,w and for all points p ∈ M.

Publié le : 2005-01-01

Zbl 1091.53009

EUDML-ID : urn:eudml:doc:282030

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     author = {Yulian T. Tsankov},
     title = {A characterization of n-dimensional hypersurfaces in $R^{n+1}$ with commuting curvature operators},
     journal = {Banach Center Publications},
     volume = {68},
     year = {2005},
     pages = {205-209},
     zbl = {1091.53009},
     language = {en},
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Yulian T. Tsankov. A characterization of n-dimensional hypersurfaces in $R^{n+1}$ with commuting curvature operators. Banach Center Publications, Tome 68 (2005) pp. 205-209. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc69-0-16/